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A pizza with diameter of 12 inches is split into 8 equally sized piece [#permalink]

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19 Sep 2017, 11:29

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A pizza with diameter of 12 inches is split into 8 equally sized pieces. (There is a figure here I cannot insert for some reason: imagine a circle with 8 points on the perimeter: A through H. The center is point "O"). 4 non-adjacent pieces are removed. What is the perimeter AOBCODEOFGOHA of the pizza now, including the inside edges of the slices?

A pizza with diameter of 12 inches is split into 8 equally sized piece [#permalink]

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19 Sep 2017, 11:46

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LakerFan24 wrote:

A pizza with diameter of 12 inches is split into 8 equally sized pieces. (There is a figure here I cannot insert for some reason: imagine a circle with 8 points on the perimeter: A through H. The center is point "O"). 4 non-adjacent pieces are removed. What is the perimeter AOBCODEOFGOHA of the pizza now, including the inside edges of the slices?

Gray areas are the remaining slices; we need their perimeter, totaled.

The pizza has had every other slice removed. Each slice has two straight edges and an arc as its perimeter.

I. Perimeter = All remaining arc lengths + all straight edges

Because the four remaining slices' perimeters are equal, find the perimeter of all four by finding the sum of the lengths four remaining arcs, and the sum of the lengths of the eight remaining straight sides.

1) Find the total length of the arcs that are left

The circumference of the pizza is \(\pi*d\), \(d = 12\), so circumference = \(12\pi\)

Half of the circumference has been removed, so the total length for the four arcs remaining is \(6\pi\)

2) Find the total of the straight edges' lengths, all of which are radii.

Radius = \(\frac{d}{2}= 6\)

There are 8 edges. If you've sketched, count. If not, each of four pieces has two straight edges, and 4 * 2 straight edges = 8 straight edges

\(8*6 = 48\) is the total of lengths of the straight edges

3) Add both totals. The perimeter is the total of the four arcs, and the total of the eight straight edges:

\(6\pi + 48\)

II. Perimeter needed = Perimeter of each remaining piece * 4

1) Find arc length of one remaining slice

The circumference of the pizza is \(\pi*d\), \(d = 12\), so circumference = \(12\pi\)

Each piece is \(\frac{1}{8}\) of the pizza, so its arc length is \(\frac{1}{8}\) of the circumference